The fallacy of causal iterative learning control

The goal of iterative learning control (ILC) is to improve the accuracy of a system that repeatedly follows a reference trajectory. This paper proves that if the ILC law is restricted to causal operators, then the ultimate ILC error can be achieved in a single trial using conventional feedback control. The feedback operator is a known function of the ILC operators alone. Hence, there is no reason to use causal ILC even if the plant is completely unknown. This equivalent feedback exists whether or not the ILC itself includes feedback. The equivalence is proved for nonlinear time-varying systems, except for the case of ILC convergence with zero error, which is proved for linear discrete-time systems